At the end of the invasion of Khwarezmia Empire, Genghis Khan chased Ala Ad-Din Muhammad to the Caspian sea, where he hid on an island and died several weeks later. However, none of the sources I can find ever mention the actual island that he died on.
I've always been curious since as far as I know there are not too many islands to speak of in the Caspian sea. The only real information I can find is that it was an Island close to the port of Abaskun.
What is the island and does it still exist today?
Well, there are two problems - the first is that we don't know where Abaskun actually is. It was purported to be in the area of the mouth of the Gorgan River, which brings us to another issue - there are numerous sandbars in the region, which come and go with the weather over a span of years. The "island" may have been a vegetated sandbar (also known as a barrier island) that has since been subsumed by the sea, after a storm or a flood or just centuries of erosion, forever lost. No-one knows exactly where the island is, or even if it still exists.
I can't find exactly where the port of Abaskun was, but it is said to be near modern Gorgun, which is near the southeast corner of the Caspian Sea.
All I see on Google Maps in that area is a couple of barrier islands (one of which technically you could claim is a peninsula). So most likely it would have been one of those. The larger one paralleling the southern coast is named Ashoorzadeh Island, the smaller one paralleling the east coast Google doesn't name.